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High-Level Proof Structure Start with path-of-sets system [C-Chuzhoy’14] Embed expander using cut-matching game of [KRV’06] Gives deg-4 sparsifier H but # of nodes in H not small New ingredient: theorem on small subgraph that preserves node-connectivity between two pairs of sets New ingredient: reduce degree to 3 by sub-sampling (non-trivial)

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Well-linked Sets A set X µ V is well-linked in G if for all A, B µ X there are min(|A|,|B|) node-disjoint A-B paths G

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Well-linked Sets A set X µ V is well-linked in G if for all A, B µ X there are min(|A|,|B|) node-disjoint A-B paths G

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Path-of-Sets System

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C1C1 C2C2 C3C3 …CrCr Each C i is a connected cluster The clusters are disjoint Every consecutive pair of clusters connected by h paths All blue paths are disjoint from each other and internally disjoint from the clusters … h

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Reducing to degree 3: idea If deg(v) = 4 delete one of the two green edges incident to it randomly Resulting graph has degree 3 v

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Reducing to degree 3: idea If deg(v) = 4 delete one of the two green edges incident to it randomly Resulting graph has degree 3 v

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Reducing to degree 3 If deg(v) = 4 delete one of the two green edges incident to it randomly Resulting graph has degree 3 Difficult part: does remaining graph have large treewidth? Embed N = £ (log k) expanders using longer path-of- sets system and cut-matching game expanders are on same set of nodes (horizontal paths) v